Kerala KEAM 5 June 2024 Solved Paper

The expression given is a form of complex numbers. Let’s simplify the expression: $$\frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right)}.$$This is a standard form for complex numbers, and by multiplying both the numerator and denominator by the conjugate of the denominator:$$\frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right)} \times \frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}.$$The denominator becomes: $$(1 + \cos\left(\frac{\pi}{5}\right))^2 + \sin^2\left(\frac{\pi}{5}\right) = 2(1 + \cos\left(\frac{\pi}{5}\right)).$$The numerator simplifies to: $$1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right),$$which is equal to: $$\cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right).$$Thus, the final expression is $\cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)$, which matches option (A).Quick Tip: The expression in the question is a standard form of a complex number in polar form. Simplifying complex expressions using conjugates is a helpful technique when dealing with complex fractions.